A "Reference Series" Method for Prediction of Properties of Long-Chain Substances

Inga Paster and Mordechai Shacham Dept. Chem. Eng.

Ben-Gurion University of the NegevBeer-Sheva, Israel

Neima BraunerSchool of Engineering Tel-Aviv University

Tel-Aviv, Israel

The NeedsPhysical property data are extensively used in chemical process design, environmental impact assessment, hazard and operability analysis, and additional applications.

Measured property values are available only for a small fraction of the chemicals used in the industry, as reactants, products or side products. Long chain substances pose special challenges, as their critical constants cannot be measured because of thermal instability.

Currently Asymptotic Behavior Correlations (ABC) are used for predicting properties of long chain substances.

ABCs represent the change of properties as nonlinear functions of nC (and/or molecular mass).

Presently Used ABC Correlations

1. Marano and Holder, Ind. Eng. Chem. Res. 36, 1887 (1997)

00 exp CC nnYYY 00, CC nnYYY

000 CC nn

Y is the property, 5 or 6 adjustable parameters ,,,,, 000 YYYnC

2. Gao et al., Fluid Phase Equilibria, 179, 207(2001)

/1

00 exp CC nnYYYY

5 adjustable parameters: ,,,,0 YY

For many homologous series only a few, inaccurate property data points are available in the low carbon number range. The use of nonlinear models with adjustable parameters based on such data for long range extrapolation is very risky and unreliable.

Property Behavior at the limit nC → ∞* Properties that approach a finite value for large carbon numbers (e.g., normal boiling and melting points, critical temperature).

Properties which are additive in nature, with a monotonic incremental change with increasing the nC. (e.g., critical volume, molar volume).

Consistency between different homologous series at the limit. The same property approaches the same value for different series.

In approaching the limit the difference between the property values for different homologs should monotonically decrease

*Marano and Holder, Ind. Eng. Chem. Res. 36, 1887 (1997)

* AIChE J, 57(2), 423–433 (2011)

Related Previous Work*

Molecular descriptors collinear with a particular property are identified based on available experimental data. From among these, the ones whose behavior at the limit nC → ∞ is similar to the property behavior are used for prediction. A linear QSPR in terms of the selected descriptor, with an optional additional correction term which exponentially decays with nC, can be developed.

Development of (linear) QSPRs with good extrapolation capabilities for high carbon number (nC) substances in homologous series.

Methodologyζy 10 Property Descriptor

The Objective of this Research

1. To establish relationships between properties of a reference series, for which the largest amount and the highest precision property data are available and the properties of a target series for which a smaller number and lower precision data points are available.

2. To use this relationship in order to determine whether the property data available for the target series is sufficient for obtaining reliable predictions.

3. To use the relationship, if the test in (2) positive, in order to predict property data for the reference series by interpolation and both short and long range extrapolations

4. Various aspects of the proposed method will be demonstrated by predicting normal boiling temperature (approaches a finite value for large carbon numbers) and critical volume (monotonic incremental change with increasing carbon number.

Ideal Gas Enthalpy of Formation (Hf) for n-alkanes and n-mercaptans

Hf decreases monotonically with increasing nC

Source: DIPPR database (Rowley et al. 2010), experimental data in bold

No. of C-atoms Value (J/kmol) Uncertainty (%) Value (J/kmol) Uncertainty (%)

3 -1.0468E+08 < 1% -6.7500E+07 < 1%4 -1.2579E+08 < 1% -8.7800E+07 < 3%5 -1.4676E+08 < 1% -1.0840E+08 < 3%6 -1.6694E+08 < 1% -1.2920E+08 < 1%7 -1.8765E+08 < 1% -1.4950E+08 < 1%8 -2.0875E+08 < 1% -1.7010E+08 < 3%9 -2.2874E+08 < 1% -1.9080E+08 < 3%

10 -2.4946E+08 < 1% -2.1090E+08 < 1%11 -2.7043E+08 < 1% -2.3250E+08 < 3%12 -2.9072E+08 < 1% -2.5320E+08 < 3%13 -3.1177E+08 < 1% - -14 -3.3244E+08 < 1% - -15 -3.5311E+08 < 1% - -16 -3.7417E+08 < 1% - -17 -3.9445E+08 < 1% - -18 -4.1512E+08 < 1% - -19 -4.3579E+08 < 1% - -20 -4.5646E+08 < 1% - -

n -alkanes n -mercaptans

-9.0000E+08

-7.0000E+08

-5.0000E+08

-3.0000E+08

-1.0000E+08

0 5 10 15 20 25 30 35 40

No. of C-atoms

Hea

t o

f F

orm

atio

n (

J/km

ol)

Predicted

DIPPR Exp

DIPPR_Pred

Modeling the Hf data of n-alkanes with the linear QSPR:

CF nH 2068000042940000-

R2 = 0.999996

The Hf of the reference series can be adequately represented as a linear function of nC for long range extrapolation

Used for model derivation

The relationship between n-Alkane and n-mercaptan, Hf data for 3 ≤ nC ≤ 12

The predicted and experimental data points are indistinguishable. The proposed relationships can be used for long range extrapolation

Prediction by linear equation rfft HH 10

β0 = (3.772 ± 0.12)E+07;

β1 = 0.9986 ± 0.006;

R2 = 0.9999

0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

2.50E+08

3.00E+08

5.00E+07 1.50E+08 2.50E+08 3.50E+08

-Hf (n-alkanes)

-Hf

(me

rca

pta

ns

)Predicted

DIPPR Exp.

Proposed by Peterson, Ind. Eng. Chem. Res., 2010 , 3492-3495

Normal Boiling Temperatures (Tb) for n-alkanes and n-alkanoic acids

No. of C-atoms Value (K) Uncertainty (%) Value (K) Uncertainty (%)

3 231.11 < 1% 414.32 < 1%4 272.65 < 1% 436.42 < 1%5 309.22 < 1% 458.95 < 1%6 341.88 < 1% 478.85 < 1%7 371.58 < 1% 496.15 < 1%8 398.83 < 1% 512.85 < 1%9 423.97 < 1% 528.75 < 1%

10 447.31 < 1% 543.15 < 1%11 469.08 < 1% 557.35 < 1%12 489.47 < 1% 571.85 < 1%13 508.62 < 1% 585.25 < 1%14 526.73 < 1% 599.35 < 1%15 543.84 < 1% 610.65 < 1%16 560.01 < 1% 623.15 < 3%17 575.30 < 1% 634.65 < 1%18 589.86 < 1% 647.15 < 1%19 603.05 < 1% 657.15 < 1%20 616.93 < 1% 668.53 < 5%

n-alkanes n-alkanoic acids

From various literature sources: ]K1091K1071[lim bn TC

Source: DIPPR database (Rowley et al. 2010), experimental data in bold

Fitting a Linear QSPR to the n-Alkane Tb data for 9 ≤ nC ≤ 20

C

C

C

C

CC n

n

n

n

nnIVDE

2log

22log

222

The linear QSPR obtained:

Tb = 917.8 (± 15.6) - 654.3(± 26.0) IVDE R2 = 0.9968

0lim IVDECn

158.917b T

The descriptor IVDE has the highest correlation with the n-alkane Tb data. This descriptor belongs to the "information indices", and it can be calculated (for the n-alkane series from:

(Requirement [1071 K – 1091 K] bT

Fitting a Linear QSPR to the n-Alkane Tb data for 9 ≤ nC ≤ 20

In this case a linear QSPR cannot be used for long range extrapolation

200

300

400

500

600

700

800

0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000

Descriptor IVDE

No

rma

l Bo

ilin

g T

em

p. (

K) Predicted

DIPPR Exp.

DIPPR Pred.

Fitting a Nonlinear QSPR to the n-Alkane Tb data for 9 ≤ nC ≤ 20

β0 = 484.7 ± 11.9; β1 = -269.3 ± 11.4; β2 = 1/(45.1 ± 0.9)

R2 = 0.999989

)]exp(1[ 21010 Cb nT

KTb 1080

300

400

500

600

700

800

0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000

Descriptor IVDE

No

rmal

Bo

ilin

g T

emp

. (K

) Predicted

DIPPR Exp

DIPPR_Pred

The nonlinear model represents both the available data and the asymptotic behavior excellently

The relationship between n-Alkane and n-Alkanoic acid Tb data for 3 ≤ nC ≤ 13

The experimental data of n-alkanoic acids is smooth, the linear relationship can be used for interpolation and short range extrap.

400

500

600

700

200 300 400 500 600 700

NBP n-alkanes (K)

NB

P a

cid

s(K

)

DIPPR Exp.

DIPPR Pred

Predicted

Prediction by linear equation rbtb TT 10

β0 = 298.96 ± 5.79;

β1 = 0.6157 ± 0.015;

R2 = 0.999

KTb 934

The relationship between n-Alkane and n-Alkanoic acid Tb data for 3 ≤ nC ≤ 13

The nonlinear relationship represents adequately the available data and converges to the correct limiting value.

Prediction by the equation:

β0 = 0.6393 ± 0.03;

β1 = 0.2459 ± 0.04;

β3 = 0.0232 ± 0.005;

R2 = 0.99989

KTbr 1080

)exp(11 3211 Cbrbrbt nTTT

400

500

600

700

200 300 400 500 600 700

NBP n-alkanes (K)

NB

P a

cid

s(K

)

DIPPR Exp.

DIPPR Pred

Predicted

Prediction of Tb for a “Target” homologous series

Plot the available Tb data of the target series versus the corresponding n-alkane data. Based on the smoothness of the curve and the number of available data points determine whether long range extrapolation is feasible.

Use nonlinear regression to obtain the coefficients of the equation:

To predict Tb for a member of the target series of a particular nC

introduce the corresponding Tb data of the n-alkane series (if available) or a predicted value obtained using the equations provided, into the above equation.

)exp(11 3211 Cbrbrbt nTTT

Special Challenges in (Long Chain) Property Prediction

Insufficient amount of property data for the reference and/or the target series

Available property data for the reference and/or the target series is too noisy.

The property value for nC → ∞* is not known

Phase change at the standard state (usually T = 298 K and P = 1 bar) appears at high nC with no corresponding property data are available. This may happen for properties specified at a standard state (For example: heat of combustion). The influence of the phase change must be considered in extrapolation.

Critical Volume (Vc) for n-alkanes and n-alkanoic acids

Vc changes monotonically with increasing nC

No. of C-atoms

Value

(m3/kmol) Uncertainty (%)

Value

(m3/kmol) Uncertainty (%)3 0.20 < 3% 0.24 < 5%4 0.26 < 3% 0.29 < 5%5 0.31 < 3% 0.35 < 3%6 0.37 < 3% 0.41 < 3%7 0.43 < 5% 0.47 < 5%8 0.49 < 5% 0.52 < 5%9 0.55 < 5% 0.58 < 10%

10 0.62 < 5% 0.64 < 10%11 0.69 < 5% 0.71 < 10%12 0.75 < 5% 0.77 < 25%13 0.82 < 10% 0.83 < 25%14 0.89 < 10% 0.89 < 25%15 0.97 < 10% 0.95 < 25%16 1.03 < 10% 1.02 < 25%17 1.10 < 25% 1.08 < 25%18 1.19 < 25% 1.14 < 25%19 1.26 < 25% 1.20 < 25%20 1.34 < 25% 1.27 < 25%

n-alkanes n-alkanoic acids

Source: DIPPR database (Rowley et al. 2010), experimental data in bold

Modeling the VC data of n-alkanes with the linear QSPR:

0

0.5

1

1.5

2

2.5

3

0 100 200 300 400 500 600 700 800

Descriptor ISIZ

Cri

tica

l V

olu

me

(m3 /k

mo

l)

Training Set

Series

Predicted

ISIZVC 0.00338770.0787111

ATAT nnISIZ 2log

Super-linear change with nC (Suggested by Marano, Gao et al.)

The relationship between n-Alkane and n-Alkene VC data for 3 ≤ nC ≤ 10

The deviation of the DIPPR pred. data can be explained by the high unceratainty (up to 25%) of these data.

Prediction by linear equation rCCt VV 10

β0 = 0;

β1 = 0.9457 ± 0.006;

R2 = 0.9995

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5

VC n-alkanes (m^3/kmol)

VC

1-a

lken

es(m

^3/

kmo

l)

DIPPR Exp.

DIPPR Pred.

Predicted

Checking the Consistency of the Available VC data for the Target Series

The plot of VCr – VCt versus VCt should yield a straight line with slope of (1- β1) for consistent data

1010 1 CrrCCrCtCr VVVVV

-0.05

-0.03

-0.01

0.01

0.03

0.05

0.1 0.2 0.3 0.4 0.5 0.6 0.7

VC n-alkanes

VC

(ref

)-V

C(t

arg

et)

1-alkenes

Acids

Slope = 1-0.946 = 0.054

Slope = unclear

Conclusions

For properties that approach a finite value for large nC a linear function of the descriptor with the highest correlation with the available data often able to provide good predictions only for interpolation and short rage extrapolation. A nonlinear expression containing the descriptor and the property value at nC → ∞*, that provides good prediction in long range extrapolation, has been developed. The linear relationship between properties of corresponding members of different homologous series, may be valid only locally. A new nonlinear relationship which holds in very wide ranges has been developed.It has been shown that the reference series approach enables optimal utilization of the available property data for checking the consistency of such data and prediction of properties in the short and long range.